diff --git a/doc/source/whatsnew/v2.0.2.rst b/doc/source/whatsnew/v2.0.2.rst
index 0d6647dc38b3c..8ef733f825d00 100644
--- a/doc/source/whatsnew/v2.0.2.rst
+++ b/doc/source/whatsnew/v2.0.2.rst
@@ -31,6 +31,7 @@ Bug fixes
 - Bug in :meth:`DataFrame.__getitem__` not preserving dtypes for :class:`MultiIndex` partial keys (:issue:`51895`)
 - Bug in :meth:`DataFrame.convert_dtypes` ignores ``convert_*`` keywords when set to False ``dtype_backend="pyarrow"`` (:issue:`52872`)
 - Bug in :meth:`Series.describe` treating pyarrow-backed timestamps and timedeltas as categorical data (:issue:`53001`)
+- Bug in :meth:`Series.rename` not making a lazy copy when Copy-on-Write is enabled when a scalar is passed to it (:issue:`52450`)
 - Bug in :meth:`pd.array` raising for ``NumPy`` array and ``pa.large_string`` or ``pa.large_binary`` (:issue:`52590`)
 
 
diff --git a/pandas/core/series.py b/pandas/core/series.py
index 84fa874831d85..e70b217e08dff 100644
--- a/pandas/core/series.py
+++ b/pandas/core/series.py
@@ -1907,7 +1907,9 @@ def to_frame(self, name: Hashable = lib.no_default) -> DataFrame:
         df = self._constructor_expanddim(mgr)
         return df.__finalize__(self, method="to_frame")
 
-    def _set_name(self, name, inplace: bool = False) -> Series:
+    def _set_name(
+        self, name, inplace: bool = False, deep: bool | None = None
+    ) -> Series:
         """
         Set the Series name.
 
@@ -1916,9 +1918,11 @@ def _set_name(self, name, inplace: bool = False) -> Series:
         name : str
         inplace : bool
             Whether to modify `self` directly or return a copy.
+        deep : bool|None, default None
+            Whether to do a deep copy, a shallow copy, or Copy on Write(None)
         """
         inplace = validate_bool_kwarg(inplace, "inplace")
-        ser = self if inplace else self.copy()
+        ser = self if inplace else self.copy(deep and not using_copy_on_write())
         ser.name = name
         return ser
 
@@ -4580,7 +4584,7 @@ def rename(
         index: Renamer | Hashable | None = None,
         *,
         axis: Axis | None = None,
-        copy: bool = True,
+        copy: bool | None = None,
         inplace: bool = False,
         level: Level | None = None,
         errors: IgnoreRaise = "ignore",
@@ -4667,7 +4671,7 @@ def rename(
                 errors=errors,
             )
         else:
-            return self._set_name(index, inplace=inplace)
+            return self._set_name(index, inplace=inplace, deep=copy)
 
     @Appender(
         """
diff --git a/pandas/tests/copy_view/test_methods.py b/pandas/tests/copy_view/test_methods.py
index 68b41682ab411..67fc91e0567ef 100644
--- a/pandas/tests/copy_view/test_methods.py
+++ b/pandas/tests/copy_view/test_methods.py
@@ -141,6 +141,7 @@ def test_methods_copy_keyword(
     "method",
     [
         lambda ser, copy: ser.rename(index={0: 100}, copy=copy),
+        lambda ser, copy: ser.rename(None, copy=copy),
         lambda ser, copy: ser.reindex(index=ser.index, copy=copy),
         lambda ser, copy: ser.reindex_like(ser, copy=copy),
         lambda ser, copy: ser.align(ser, copy=copy)[0],
@@ -158,6 +159,7 @@ def test_methods_copy_keyword(
         lambda ser, copy: ser.set_flags(allows_duplicate_labels=False, copy=copy),
     ],
     ids=[
+        "rename (dict)",
         "rename",
         "reindex",
         "reindex_like",
diff --git a/pandas/tests/frame/indexing/test_setitem.py b/pandas/tests/frame/indexing/test_setitem.py
index af3632bffe948..3edfd47cb05a1 100644
--- a/pandas/tests/frame/indexing/test_setitem.py
+++ b/pandas/tests/frame/indexing/test_setitem.py
@@ -316,7 +316,7 @@ def test_frame_setitem_existing_datetime64_col_other_units(self, unit):
         df["dates"] = vals
         assert (df["dates"].values == ex_vals).all()
 
-    def test_setitem_dt64tz(self, timezone_frame):
+    def test_setitem_dt64tz(self, timezone_frame, using_copy_on_write):
         df = timezone_frame
         idx = df["B"].rename("foo")
 
@@ -331,12 +331,16 @@ def test_setitem_dt64tz(self, timezone_frame):
 
         # assert that A & C are not sharing the same base (e.g. they
         # are copies)
+        # Note: This does not hold with Copy on Write (because of lazy copying)
         v1 = df._mgr.arrays[1]
         v2 = df._mgr.arrays[2]
         tm.assert_extension_array_equal(v1, v2)
         v1base = v1._ndarray.base
         v2base = v2._ndarray.base
-        assert v1base is None or (id(v1base) != id(v2base))
+        if not using_copy_on_write:
+            assert v1base is None or (id(v1base) != id(v2base))
+        else:
+            assert id(v1base) == id(v2base)
 
         # with nan
         df2 = df.copy()